scholarly journals Accuracy of reconstruction of the multidimensional probability density by wavelet estimates of one-dimensional projections

2018 ◽  
pp. 21-30
Author(s):  
Artem Borisov ◽  
◽  
Oleg Shestakov ◽  
◽  
Keyword(s):  
Probability Density ◽  
One Dimensional ◽  
Multidimensional Probability

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

scholarly journals A fractional generalization of the dirichlet distribution and related distributions

2021 ◽  
Vol 24 (1) ◽  
pp. 112-136
Author(s):  
Elvira Di Nardo ◽  
Federico Polito ◽  
Enrico Scalas
Keyword(s):  
Monte Carlo ◽  
Probability Density Function ◽  
Monte Carlo Simulations ◽  
Probability Density ◽  
Dirichlet Distribution ◽  
One Dimensional ◽  
Reasonable Approximation ◽  
Generalized Dirichlet Distribution ◽  
The One ◽  
Generalized Dirichlet

Abstract This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the n partitions of the interval [0, Wn ] are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (2) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

The nature of the electronic states in disordered one-dimensional systems

1963 ◽  
Vol 274 (1359) ◽  
pp. 529-545 ◽  
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Analytical Calculation ◽  
Homogeneous Boundary Condition ◽  
High Electron ◽  
One Dimensional ◽  
Computer Calculations ◽  
A Chain ◽  
Zero Potential

We consider an electron moving in the field of a one-dimensional infinite chain of identical potentials separated by regions of zero potential, the lengths s of these regions being distributed according to a probability density function p(8) . If we define the reduced phase of a real solution of the wave equation as the principal value of arctan ( — ψ'/kψ ) and є i as the reduced phase at the point x i immediately to the left of the i th atomic potential, it is shown for all bounded p(s) and sufficiently high electron energies that the є i are distributed according to a probability density function which depends on the direction of integration from a specified homogeneous boundary condition. This result is shown to imply that the eigenfunctions for such systems are localized in the sense that the envelope of such a function decays on average in an exponential manner on either side of some region. An analytical calculation for a random chain of δ-functions gives the decay of the nodes explicitly for high energies, and numerical calculations of the decay for a liquid model are presented. Further support for the theory is provided by computer calculations of some of the eigenfunctions of a chain of 1000 randomly placed δ-functions.

scholarly journals MEAN-FIELD CRITICAL BEHAVIOR AND ERGODICITY BREAK IN A NONEQUILIBRIUM ONE-DIMENSIONAL RSOS GROWTH MODEL

2012 ◽  
Vol 23 (03) ◽  
pp. 1250019 ◽  
Author(s):  
J. RICARDO G. MENDONÇA
Keyword(s):  
Phase Transition ◽  
Probability Density ◽  
Critical Behavior ◽  
Random Number Generator ◽  
Mean Field ◽  
System Size ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Roughening Transition ◽  
Pseudo Random Number Generator

We investigate the nonequilibrium roughening transition of a one-dimensional restricted solid-on-solid model by directly sampling the stationary probability density of a suitable order parameter as the surface adsorption rate varies. The shapes of the probability density histograms suggest a typical Ginzburg–Landau scenario for the phase transition of the model, and estimates of the "magnetic" exponent seem to confirm its mean-field critical behavior. We also found that the flipping times between the metastable phases of the model scale exponentially with the system size, signaling the breaking of ergodicity in the thermodynamic limit. Incidentally, we discovered that a closely related model not considered before also displays a phase transition with the same critical behavior as the original model. Our results support the usefulness of off-critical histogram techniques in the investigation of nonequilibrium phase transitions. We also briefly discuss in the appendix a good and simple pseudo-random number generator used in our simulations.

scholarly journals ONE DIMENSIONAL BROWNIAN MOTION WITH HOLDING AND JUMPING BOUNDARY

2022 ◽  
Author(s):  
Yuk Leung
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Probability Density ◽  
Boundary Problem ◽  
Exponential Distribution ◽  
Boundary Point ◽  
Nonlocal Boundary ◽  
Unit Interval ◽  
One Dimensional ◽  
Nonlocal Boundary Problem

Let a particle start at some point in the unit interval I := [0, 1] and undergo Brownian motion in I until it hits one of the end points. At this instant the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I with a probability density μ0 or μ1 parametrized by the boundary point it was from. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper [10]. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process.

Estimation of the integral of the square of derivatives of symmetric probability densities of one-dimensional random variables

Metrologiya ◽  
2020 ◽  
pp. 15-27
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Standard Deviation ◽  
Probability Density ◽  
Random Variables ◽  
Random Variable ◽  
Density Estimates ◽  
One Dimensional ◽  
Probability Densities ◽  
The One ◽  
Derivatives Of ◽  
Selection Of

When substantiating the method of fast selection of the bandwidth of kernel probability density estimates, a constant was found that is a functional of the second density derivative. In this paper, the obtained result is generalized to derivatives of symmetric probability densities of different orders. The functional dependences of the constants under study on the coeffi cient of antikurtosis of a random variable are established. The regularities peculiar to them are investigated. Based on the results obtained, a method for estimating functionals from derived probability densities has been developed, which involves the following actions. In the original sample estimated standard deviation of the one-dimensional random variables and the coeffi cient of antikurtosis. Using the reconstructed functional dependences on the antikurtosis coeffi cient, the constants are estimated, which are functionals of the derivatives of the probability density. With known estimates of the standard deviation of the investigated random variable and the considered constant, the values of the functional from the derivative of the probability density of the selected order are calculated. The obtained results are confi rmed by the analysis of the data of computational experiments. It is established that with increasing order of the derivative, the values of the estimates of the studied functionals increase. This fact is explained by the complication of the integrand function in the considered functionals. The proposed method provides objective results for the fi rst three derivatives of the probability density of a random variable. The obtained conclusions are confi rmed by the results of the confi dence estimation of the investigated functionals.

Estimating the integral of the square of the probability density of a one-dimensional random variable

2020 ◽  
pp. 22-28
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Probability Density ◽  
Computational Experiment ◽  
Variable Interval ◽  
Random Variable ◽  
Optimal Number ◽  
Functional Dependencies ◽  
Distribution Law ◽  
Generalized Model ◽  
One Dimensional ◽  
Probability Densities

When choosing the optimal number of sampling intervals of the range of values of a one-dimensional random variable, it is established that the functional of the square of the probability density is a constant. The values of the constant are independent of the probability density parameters. The functional dependencies of the studied constant on the coeffi cient of antikurtosis of the distribution law of a random variable are determined. The analysis of the established dependences for families of lognormal probability densities, Student's distribution laws and families of probability densities with Gauss distribution is carried out. Based on the results obtained, a generalized model is formed between the studied constant and the antikurtosis coeffi cient. The generalized model does not depend on the type of probability density, but is determined by the estimation of the antikurtosis coeffi cient. On this basis, we develop a method for estimating the integral of the square of the probability density, which involves the following actions. The random variable interval and the antikurtosis coeffi cient are estimated from the initial sample. At known values of these estimates, the integral of the square of the probability density is calculated. The eff ectiveness of the proposed method is confi rmed by the results of computational experiments. The conditions of the computational experiment diff er signifi cantly from the information used in the synthesis of models of the dependence between the studied constant and the antikurtosis coeffi cient. The conditions of competence of the method of estimating the probability density square integral from the antikurtosis coeffi cient are established using the proposed models of the dependence of the studied constant on the conditions of the computational experiment.

Sign in / Sign up

Export Citation Format

Share Document